Good Questions: How Should We Cover this Room's Floor?

New place, and hopefully some new furniture (electric fireplace will not be staying). This room will serve three purposes; a living/tv room, a dining area, and a spot for a small desk w/computer. The dimensions are 20 x 13, and I'm not sure what to do about floor coverings...

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I'd like to buy an area rug that is 10 x 12 for the back half of the room for our living area (near the terrace doors), with a sofa along the long wall on the left. But is it okay to leave the floors bare at the front half of the room, where our dining table will be to the left and our desk area to the right?

Or must we define each area with a separate rug (which just seems crazy to me)? Or would an area rug or perhaps a bound piece of carpet covering the majority of the floor be more appropriate, say around 18 x 12?